Bulk viscosity of CO2 from Rayleigh-Brillouin light
scattering spectroscopy at 532 nm
Wang, Yuanqing; Ubachs, Wim; Van De Water, Willem
Publication date
2019
Document Version
Final published version
Published in
Journal of Chemical Physics
Citation (APA)
Wang, Y., Ubachs, W., & Van De Water, W. (2019). Bulk viscosity of CO2 from Rayleigh-Brillouin light
scattering spectroscopy at 532 nm. Journal of Chemical Physics, 150(15), [154502].
https://aip.scitation.org/doi/full/10.1063/1.5093541
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Brillouin light scattering spectroscopy at
532 nm
Cite as: J. Chem. Phys. 150, 154502 (2019); https://doi.org/10.1063/1.5093541
Submitted: 21 February 2019 . Accepted: 17 March 2019 . Published Online: 17 April 2019 Yuanqing Wang , Wim Ubachs , and Willem van de Water
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Bulk viscosity of CO
2
from Rayleigh-Brillouin
light scattering spectroscopy at 532 nm
Cite as: J. Chem. Phys. 150, 154502 (2019);doi: 10.1063/1.5093541Submitted: 21 February 2019 • Accepted: 17 March 2019 • Published Online: 17 April 2019
Yuanqing Wang,1 Wim Ubachs,1,a) and Willem van de Water2 AFFILIATIONS
1Department of Physics and Astronomy, LaserLaB, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands 2Laboratory for Aero and Hydrodynamics, Faculty of Mechanical, Maritime and Materials Engineering,
Delft University of Technology, Leeghwaterstraat 29, 2628CB Delft, The Netherlands
a)
Electronic mail:w.m.g.ubachs@vu.nl
ABSTRACT
Rayleigh-Brillouin scattering spectra of CO2were measured at pressures ranging from 0.5 to 4 bars and temperatures from 257 to 355 K
using green laser light (wavelength 532 nm, scattering angle of 55.7○). These spectra were compared to two line shape models, which take
the bulk viscosity as a parameter. One model applies to the kinetic regime, i.e., low pressures, while the second model uses the continuum, hydrodynamic approach and takes the rotational relaxation time as a parameter, which translates into the bulk viscosity. We do not find a significant dependence of the bulk viscosity with pressure or temperature. At pressures where both models apply, we find a consistent value of the ratio of bulk viscosity over shear viscosityηb/ηs= 0.41± 0.10. This value is four orders of magnitude smaller than the common value
that is based on the damping of ultrasound and signifies that in light scattering only relaxation of rotational modes matters, while vibrational modes remain “frozen.”
Published under license by AIP Publishing.https://doi.org/10.1063/1.5093541
I. INTRODUCTION
The light scattering properties of carbon dioxide remain of interest, both from a fundamental perspective studying the relax-ation in molecular gases and for determining their thermody-namic properties, as well as from an applied perspective. Details of Rayleigh-Brillouin (RB) phenomena, scattering spectral profiles of CO2 gas at differing pressures and temperatures,1–3 as well as its
cross section,4are of relevance for current and future remote sens-ing exploration of the planetary atmospheres where CO2is the main
constituent, either under high-pressure and high-temperature con-ditions as on Venus5 or under low pressure and low-temperature conditions as on Mars.6 The fact that carbon dioxide is the prime greenhouse gas has spurred large-scale activity in the transforma-tion of this gaseous species,7,8through catalytic hydrogenation,9and electrochemical conversion into renewable energy,10 as well as in plasma-driven dissociation for the synthesis of fuel from CO2.11,12
Apart from issues of capture, fixation, and transformation of tera-tons of carbon dioxide, its storage and transport, either in the liq-uid or gas phase, forms an important challenge.13,14For these pur-poses, the study of the transport coefficients of CO2gas, such as
thermal conductivity,15heat capacity, and shear viscosity,16,17is of practical importance. Light scattering is an elegant way to deter-mine the thermodynamic properties of a gas because the Rayleigh scattering phenomenon resulting in the elastic peak is connected to entropy fluctuations,18while the Brillouin-side peaks are associ-ated with density fluctuations or sound.19,20 The macroscopic gas transport coefficients govern the scattering spectral profiles and can in turn be deduced from the measurement of such profiles.21 This holds for both spontaneous RB-scattering22–24 and coherent RB-scattering.25,26
The bulk viscosity,ηb,27,28is the most elusive transport
coeffi-cient. It is associated with the relaxation of internal degrees of free-dom of the molecule, i.e., rotations and vibrations. The bulk viscosity is commonly measured from the damping of ultrasound at frequen-cies in the megahertz domain.29It can also be retrieved from the light scattering spectrum of molecular gases. This was demonstrated by Panet al.2and recently for N2, O2, and air by Gu and Ubachs30and
for N2O gas by Wanget al.31In light scattering, the frequenciesfs
involved are those of sound with wavelengths comparable to that of light, three orders of magnitude larger than the frequencies used to measureηbfrom ultrasound experiments. The bulk viscosity of CO2
from light scattering is found to be four orders of magnitude smaller than that from ultrasound experiments.1–3A simple explanation is that at high (hypersound) frequencies the relaxation of vibrational modes of the CO2molecule no longer plays a role, i.e., the vibrational
energy stays frozen in.
The bulk viscosity can be expressed in terms of relaxation times of intramolecular degrees of freedom
ηb= p
2 (3 + ∑iNi)2∑i
Niτi, (1)
wherep is the pressure, Niis the number of degrees of freedom of
modei (rotations, vibrations), and τiis the relaxation time (τrot,
τvib).32A frequency-dependent version of this formula, depending
on the productfsτi, was given by Meijeret al.26Carbon dioxide is a
linear molecule with 2 rotational degrees of freedom. In the case of frozen vibrations (fsτvib≫ 1), Eq.(1)reduces to
ηb=
4
25p τrot. (2) The vibrational relaxation time strongly decreases with temperature, and so does the bulk viscosity. In a simple model, Landau and Teller proposed an exponential dependence of the chance of relaxation on the ratio of the collision interaction time and the vibration period.29 This leads to a scaling prediction for the temperature dependence of the bulk viscosity which agrees with the experiment.27On the other hand, a classical analysis of collisions of rigid rotators by Parker33 results in a scaling expression forτrotwhich predicts an “increase”
inτrotwith increasing temperature.
The bulk viscosity changes with temperature motivated the present study in which laser-based light scattering measurements in CO2gas are carried out in a pressure regime of 0.5–4 bars and
in a temperature regime of 257–355 K. Accurate, highly spectrally resolved and high signal-to-noise scattering line profiles are mea-sured at a scattering angle of 55.7○ and a scattering wavelength of
λi= 532 nm, a wavelength commonly used in Lidar applications.
Under these conditions of a longer wavelength and a smaller scatter-ing angle than in a previous study,3the Brillouin-side peaks become more pronounced in the scattering spectrum.
Experimental data are analyzed in the context of two models for the spectral line shape. One model applies at low pressures, the kinetic regime where the mean free path between collisions is com-parable to the wavelength of light,34,35while the other one is valid in
the hydrodynamic regime.36Values of the bulk viscosity are deter-mined using a least squares method by comparing model spectra to the measured ones.
The remainder of this article consists of an experimental sec-tion, a section discussing the bulk viscosity and model descriptions for RB-scattering, a presentation of results in the context of appli-cable models to describe the scattering spectrum, followed by a conclusion.
II. EXPERIMENT
The experimental apparatus for measuring the RB-scattering spectral profiles of CO2 is displayed in Fig. 1. The laser source
provides continuous wave radiation atλi= 532.22 nm at a
band-width of less than 5 MHz. RB-scattered light is produced from the laser beam of 5 W intensity traversing a gas cell equipped with a gas inlet valve and a pressure sensor. Brewster-angled windows are mounted at the entrance and exit ports, and black paint covers the inside walls, to reduce unwanted scatter contributions. A temper-ature control system equipped with Peltier elements is employed for heating, cooling, and keeping the cell at a constant tempera-ture with uncertainty less than 0.1 ○C. RB-scattered light is
cap-tured at a scattering angle θ = (55.7 ± 0.3)○ defined by the
set-ting of a slit in the scatter beam path (seeFig. 1), which also lim-its the opening angle for collecting RB-scattered light to less than 0.5○. The exact scatter angle and uncertainty are determined with
FIG. 1. Schematic of the experimental apparatus. A Verdi-V5 laser provides continuous wave light at 532.22 nm at a power of 5 W and a bandwidth less than 5 MHz. The laser light is split into two beams: The pump beam crosses the RB-scattering gas cell producing scattered light that is captured under an angleθ = (55.7 ± 0.3)○. The small fraction reference beam transmitted through M1is used to align the beam path after the gas cell toward the detector. The scattered light is analyzed in a Fabry-Pérot Interferometer (FPI), with a free spectral range of 2.9964 GHz and an instrument linewidth of (58 ± 3) MHz, and is collected on a photomultiplier tube (PMT). Mirrors, lenses, and diaphragm pinholes are indicated as Mi, Li, and Di, respectively. A slit of 500µm is inserted to limit the opening angle for collected scattering light, therewith optimizing the resolution.
FIG. 2. (a) Experimental data for the RB-light scattering spectrum of CO2 mea-sured at λi = 532 nm, θ = 55.7○, and (p,T) conditions as indicated, cor-responding to a uniformity parameter
y = 1.88; the spectrum on the right shows
an enlargement of the central part indi-cated in gray. (b) Residuals from com-parison with the Tenti-S6 model by using thermal conductivityλthfrom Eq.(3)and a fitted value forηb. (c) Residuals with λthfrom the work of Uribe et al.15
a rotatable stage operated as a goniometer. A reference beam as depicted inFig. 1is used for aligning the collection and detection system. The scattering angle determines the scattered light wave vectorksc
ksc=
2π λi
2n sin(θ/2), withn being the index of refraction.
The scattered light propagates through a bandpass filter (Mate-rion, T > 90% at λi = 532 nm, bandwidth △λ = 2.0 nm) onto
a Fabry-Pérot interferometer (FPI) via an optical projection sys-tem consisting of a number of lenses and pinholes to reduce stray light and contributions from Raman scattering. Finally, the scattered photons are detected on a photomultiplier tube (PMT), processed, and stored in a data acquisition system. The FPI is half-confocal, the curved mirror having a radius of curvature ofr =−12.5 mm. Mirror reflectivities are 99%. The FPI has an effective free spectral range (FSR) of 2.9964 GHz, which is determined through frequency-scanning a laser (a narrowband tunable cw-ring dye laser) over more than 1000 modes of the FPI while measuring the laser wavelength by a wavelength meter (Toptica HighFinesse WSU-30), hence yield-ing an uncertainty in the FSR below 1 MHz. The instrument width, yielding a value ofσνinstr= 58.0± 3.0 MHz (FWHM), is determined
by using the reference beam while scanning the piezo-actuated FPI, following the methods discussed by Gu et al.37 It includes the bandwidth of the incident laser. The instrument function is verified to exhibit the functional form of an Airy function, which may be well approximated by a Lorentzian function during data analyses.
RB-scattering spectral profiles were recorded by piezo-scanning the FPI at integration times of 1 s for each step, usually over 18 MHz. Typical detection rates were∼2000 count/s for con-ditions of 1 bar pressure. A full spectrum covering lots of consec-utive RB-peaks and 10 000 data points was obtained in about 3 h. The piezo-voltage scans were linearized and converted to frequency scale by fitting the RB-peak separations to the calibrated FSR-value. The linearization procedure also corrects for frequency drifts of the laser, which were measured to amount to 10–100 MHz/h, depend-ing on temperature drifts in the laboratory. Finally, a collocated spectrum was obtained by cutting and adding all individual record-ings over∼60 RB-peaks.37 In a final step, the RB scattering files were averaged to improve the signal to noise ratio. This pro-cedure yields a noise level of∼0.4% (with respect to peak height) for the 1 bar pressure case. A single typical light scattering spec-trum recorded at 1 bar and room temperature, measured in a typ-ical recording time of∼3 h, is displayed inFig. 2.Figure 2and its inset demonstrate the signal-to-noise ratio attainable in the present setup.
III. RB-SCATTERING AND LINE SHAPE MODELS
In light scattering, the key quantity is the uniformity parameter y, which—up to a constant—is defined as the ratio of the scattering wavelength over the mean free path between collisions, which can be shown to equal
y= p kscv0ηs
with thermal velocity v0 = (2kBT/m)1/2, wherekBis the
Boltz-mann constant, m is the molecular mass, and ηs is the shear
viscosity.
Valuesy= O(1) pertain to the kinetic regime, and models must be based on the Boltzmann equation. There, spectra do not devi-ate strongly from the Rayleigh (Maxwellian) line shape. At larger values ofy, many mean free paths fit in a wavelength and a hydro-dynamic, continuum approach applies. The Brillouin-side features become more and more prominent with increasingy and occur at frequency shiftsfs =±υsksc/2π, with υs being the speed of sound.
Our data are in the intervaly = [0.7–9] and, therefore, range from the kinetic into the hydrodynamic regime.
A. The Tenti model
The Tenti model, originally developed for analyzing RB-scattering in molecular hydrogen and diatomic molecules,34,35is a widely used model for light scattering spectra in the kinetic regime. It uses the Wang Chang–Uhlenbeck eigentheory which takes known values of the macroscopic transport coefficients as input.38 This input consist of values for the shear viscosityηs, thermal
conduc-tivityλth, the molar heat capacityCintof internal modes of motion
(rotations, vibrations), and the bulk viscosityηb. The model agrees
well with experiments,3,24,25where it was established that the six-mode version of the Tenti six-model (hereafter called the Tenti-S6
TABLE I. Datasets for RB-scattering measurements in CO2gas recorded under conditions as indicated. The uniformity parameter is y. For values of the temperature-dependentηsandλth, we used the values Boushehri et al.39and Eq.(3), respectively. The bulk viscosity and the ratiosηb/ηsare derived in a fit to the experimental data. The bulk viscosityηT
bis based on the Tenti-S6 model, whileηH
b is based on the Hammond–Wiggins hydrodynamic model. The parameter Cint= 2/2
R, for all cases, is the heat capacity of rotational motion.
Dataset p T ηs(× 10−5) λth(× 10−3) ηTb(× 10−5) ηTb/ηs ηHb(× 10−5) ηHb/ηs y Unit bars K Pa s W/m k Pa s Pa s 0.5 bar 0.500 273.2 1.37 13.4 0.64 0.47 1.03 0.500 293.2 1.47 14.3 0.72 0.49 0.93 0.505 313.2 1.56 15.3 0.91 0.58 0.85 0.508 333.2 1.66 16.2 1.65 0.99 0.78 0.503 353.2 1.75 17.1 1.79 1.02 0.71 1 bar 1.033 258.1 1.30 12.7 0.50 0.38 0.30 0.23 2.32 1.038 274.3 1.38 13.4 0.46 0.33 0.24 0.18 2.13 1.011 293.2 1.47 14.3 0.54 0.37 0.28 0.19 1.88 1.055 312.9 1.56 15.2 0.69 0.45 0.31 0.20 1.79 1.048 330.8 1.65 16.1 0.77 0.47 0.33 0.20 1.62 1.028 353.2 1.75 17.1 0.67 0.38 0.34 0.19 1.46 2 bars 2.012 257.4 1.29 12.6 0.47 0.37 0.49 0.38 4.54 2.037 274.5 1.38 13.4 0.46 0.34 0.45 0.33 4.18 2.000 293.2 1.47 14.3 0.40 0.27 0.37 0.25 3.71 2.047 312.9 1.56 15.2 0.60 0.38 0.47 0.30 3.47 2.050 331.8 1.65 16.1 0.70 0.42 0.51 0.31 3.15 2.042 354.8 1.76 17.1 0.64 0.36 0.40 0.23 2.89 3 bars 3.012 257.1 1.29 12.6 0.66 0.51 0.73 0.57 6.80 2.996 273.2 1.37 13.4 0.65 0.47 0.63 0.46 6.18 3.037 295.7 1.48 14.4 0.66 0.45 0.68 0.46 5.59 3.050 313.7 1.56 15.3 0.77 0.49 0.75 0.48 5.15 3.064 332.4 1.65 16.1 0.98 0.59 0.89 0.54 4.76 3.021 354.4 1.76 17.1 0.75 0.43 0.61 0.35 4.28 4 bars 4.026 258.1 1.29 12.6 0.98 0.76 1.07 0.83 9.04 4.052 274.9 1.34 13.5 0.70 0.51 0.79 0.57 8.29 4.048 295.2 1.48 14.4 0.61 0.41 0.70 0.47 7.46 4.041 313.1 1.56 15.3 0.73 0.47 0.78 0.50 6.84 4.042 332.7 1.65 16.2 0.86 0.52 0.86 0.52 6.27 4.000 353.5 1.75 17.1 0.94 0.54 0.89 0.51 5.68
model) yields a better agreement with experiments than the seven-mode variant.
We use the values for a temperature dependent shear viscosity ηs(T) for CO2from the work of Boushehriet al.39Of the transport
coefficients needed in the model,λth,Cint, andηbdepend on the
par-ticipation of intramolecular modes of motion. We assume that at the frequencies associated with light scattering, only rotations partici-pate in the exchange of internal and kinetic energy so thatCintfor
the linear CO2molecule becomesCint= 2/2R with R being the gas
constant.
For the thermal conductivity λth, Uribe et al.15 listed
temperature-dependent values. As in the present study only the thermal conductivity associated with “rotational” relaxation is con-sidered, these values are not straightforwardly applicable to RB-scattering data. We need a value that reflects rotational internal energy only. For polyatomic gases, a high-frequency value forλth
was estimated from the Eucken relation, which expressesλthas a
function of the shear viscosityηs, the diffusivity D, and the heat
capacityCintof internal motion
λth=52ηsCt+ρ D Cint, (3)
with Ct = 3/2 R, the heat capacity of kinetic motion, Cint = 2/2
R, and the temperature-dependent diffusivity D taken from the work of Boushehriet al.39 At temperatureT = 296.55 K, the low-frequency value isλth= 1.651× 10−2W/m K (Ref.15), whereas
Eq.(3)predicts a high-frequency valueλth= 1.452× 10−2W/m K.
As will be demonstrated below, the smaller value indeed pro-duces a better fit of the kinetic model. The bulk viscosityηb, our
prime quantity of interest, is determined using a least squares procedure.
B. The Hammond–Wiggins model
At the other end of the uniformity scale, we seek confronta-tion with a hydrodynamic, continuum model by Hammond and
FIG. 3. Experimental data for RB-scattering in CO2as measured for the various pressure and temperature conditions as indicated. The data are on a scale of normalized integrated intensity over one FSR. The 29 spectra pertain to the entries inTable I.
Wiggins.36 Unlike the Tenti model, which is built on (tensorial) eigenvectors of the linearized collision operator, the hydrodynamic model is built on (tensorial) moments of the space-time distri-bution function, i.e., hydrodynamic quantities. The hydrodynamic model takes the shear viscosityηs, the diffusivityD, and the heat
capacity of internal motionCintas parameters. The rotational
relax-ation timeτrotis determined using a least squares procedure, from
which the bulk viscosityηb is computed using Eq.(2). Allowance
for rotational relaxation only is done through the choice of Cint= 2/2R.
The evaluation of both kinetic and hydrodynamic models can be done extremely quickly. Where their range of validity overlaps, the derived values of the bulk viscosity should agree.
IV. RESULTS AND DISCUSSION
In this section, we will first present experimental data on light scattering in CO2, followed by an analysis in terms of two
complementary spectral line models. We finally summarize the results of the temperature-dependent bulk viscosity.
A. Measurements: Light scattering in CO2
Measurements of the RB-scattering spectral profile of CO2gas
were performed for conditions of 0.5–4 bars pressure and temper-atures in the range between 258 and 355 K, as listed inTable I. In
Table I, the accurately measuredp and T conditions for 29 (p, T) measurement combinations as well as the temperature-dependent transport coefficients are listed: shear viscosityηsand thermal
con-ductivityλth. For all measurements, a value for the internal
molecu-lar heat capacity ofCint= 2/2R is adopted.
InFig. 3, the RB light spectra for the 29 different (p,T) com-binations are graphically displayed. A qualitative inspection shows, when comparing profiles from the top-row down, the pressurep is increased and therewith the y-parameter is increased, and hence, the spectra show more pronounced Brillouin-side peaks. Indeed, at higher uniformity parameters y, the hydrodynamic regime is
FIG. 4. Plot of calculated residuals between experimental data and spectral profiles obtained from a fit to the Tenti-S6 model with transport coefficients as listed inTable Iand optimized values forηbas derived in the fit. Note the one-to-one correspondence with the 29 graphs of spectra inFig. 3.
approached resulting in well-isolated acoustic side modes. Simi-larly, while going from left to right along the columns, the tempera-tureT is increased, associated with a lowering of the y-parameter, and hence, the Brillouin-side peaks become less pronounced. In the following, the experimental profiles will be compared to the Tenti-S6 model and the Hammond–Wiggins hydrodynamic model.
B. Comparison with the two models
For a quantitative analysis of the data, a comparison is made with the Tenti-S6 model, which was developed into a code25,40that was included in fitting routines for analyzing both spontaneous and coherent RB-scattering.24,26 In comparing model and experi-ments, the bulk viscosityηbwas determined using a least-squares
procedure, minimizing the mean squared deviation
χ2= 1 N N ∑ i=1 [Ie(fi) − Im(fi)]2 δ2(fi) ,
whereIe(fi) andIm(fi) are the experimental and modeled amplitude
of the spectrum at (discrete) frequencyfi, andN is total number of
the experimental data. The errorδ(fi) ofIe(fi) is estimated as the
square root of the number of collected photons.
First, an analysis is made of the spectrum presented inFig. 2. Least-squares fits are performed for determining the value of the bulk viscosity ηb, invoking two values for the thermal
conductiv-ity: firstλth= 14.3 mW/m K as resulting from the modified Eucken
approach and second the value obtained from a direct measurement at acoustic frequenciesλth= 16.2 mW/m K.39The residuals
plot-ted inFig. 2show that in the first approach a peak residual of 3% is found, where in the latter approach the peak deviation amounts to 5%. This supports the validity of the treatment of thermal
FIG. 5. Plot of calculated residuals between experimental data and spectral profiles obtained from a fit to the Hammond-Wiggins model with transport coefficients as listed in
Table Ifrom 1 bar to 4 bars. The optimized values forτrotas derived in the fit and the correspondingηHb as derived using Eq.(2)when setting the vibrational relaxational time
FIG. 6. Summary of results: the ratio of bulk to shear viscosityηb/ηsof CO2as a function of temperature T. Open circles: as estimated from the experimental data using the Tenti-S6 model; closed dots: using the Hammond–Wiggings hydrody-namic model. Red line in (c): Landau– Teller theory “scaled down by a factor” 104. Blue dashed line in (c): prediction of Parker model33for rotational relaxation. Violet bars in (c) and (d): estimate of meanηb/ηs.
conductivity following the modified Eucken relation in this study conducted at hypersound frequencies. In such an approach focus-ing on hypersound, the internal heat capacity is set atCint= 2/2R,
signifying that two rotational degrees of freedom are involved and vibrational relaxation is “frozen.”
Least-squares fitting procedures based on the Tenti-S6 model were applied to the large body of 29 datasets on CO2for (p, T)
values, as displayed in Fig. 3. With inclusion of values for the transport coefficients as listed inTable I, optimized values forηTb
were derived. The resulting values are listed in Table I. Based on these fits and optimized ηTb values, residuals between
exper-imental data and the Tenti-S6 model description are calculated and displayed in Fig. 4. These residuals provide insight into the quality of the fit, its accuracy, and the applicability of the model.
Similarly, the hydrodynamic model is used to compare the experimental and model spectra for the data of pressures above 1 bar. Here, the rotational relational timeτrotwas adopted as a free
param-eter, from which the bulk viscosity was calculated using Eq. (2).
Figure 5displays the residual between the experimental data and this model.
C. Bulk viscosities
Our main result, the ratioηb/ηs as a function of temperature
and pressure is summarized inFig. 6. Atp = 0.5 bar, the spectral line shape is not very sensitive to variation of ηb; at the highest
pressure, the models deviate significantly from the experiment. At p = 1 bar (corresponding to y = [1.4–2.3]), the hydrodynamic model does not yet apply, and the values ofηb/ηsof the two models differ
significantly.
Figure 6also shows the prediction of the Landau–Teller scaling, which captures low-frequency experimental data.27However, with a crucial proviso: “it is scaled down by a factor” 104. This vividly illustrates the dramatic effect of high frequencies on the ratioηb/ηs.
At these high frequencies, only rotational relaxation remains. Based on the analysis of classical trajectories, Parker33 derived a scaling
expression for the ratioηb/ηs
ηb/ηs∝⎡⎢⎢⎢ ⎢⎣1 + π3/2 2 ( T∗ T ) 1/2 +(π 2 4 +π) T∗ T ⎤⎥ ⎥⎥ ⎥⎦ −1 withT∗
= 82.6 K, the temperature associated with the well depth of the O–O interaction potential.41 This prediction, scaled on ηb/ηs= 0.5 atT = 250 K, is also shown inFig. 6.
We find no significant dependency on pressure or temperature. At pressures where both kinetic and hydrodynamic models apply, we find an averageηb/ηs= 0.33± 0.06 at p = 2 bars and ηb/ηs= 0.48
± 0.06 at p = 3 bars. These averages, together with their uncertainty, are also indicated in Fig. 6. Our present numbers are consistent with the finding obtained from the light scattering experiments on CO2in the UV-range (λ = 366.8 nm) covering the parameter space
y = [0.9–3.7], and yielding ηb= (5.7± 0.6) × 10−6Pa s,3which gives
rise toηb/ηs = 0.39± 0.04. These values should be compared to
ηb= 4.6× 10−6Pa s (fory = [3.3–8.2]) by Lao et al.,1
correspond-ing toηb/ηs= 0.31, andηb= 3.7× 10−6Pa s fory = [0.44–3.54] by
Panet al.,2for which the ratioηb/ηs= 0.25.
V. DISCUSSION AND CONCLUSION
In this paper, we study Rayleigh-Brillouin scattering over a range of pressures with the aim of determining the bulk viscos-ity using two different types of models for the spectral line shape. Where the range of applicability of these two models overlaps, we find consistent values of the bulk viscosity.
At low frequencies, the bulk viscosity depends strongly on tem-perature, which is caused by the temperature dependence of the vibrational relaxation rate. We do not find a significant temperature dependence, not even the one predicted for the increase in the bulk viscosity with the temperature due to the increase in the rotational relaxation time. We findηb/ηs= 0.41± 0.10 at pressures of 2 and 3
For N2, a measurement of sound absorption at low pressure
yielded a value of the bulk viscosity, expressed relative to the shear viscosity asηb/ηs= 0.73 (Ref.42). Cramer27showed that this ratio
should increase from 0.4 to 1 as the temperature changes from 100 K to 420 K. Indeed, Gu and Ubachs43 experimentally determined a ratio ofηb/ηs= [0.46–1.01] from RB-scattering for the temperature
range 254.7–336.6 K. Hence, for measurements at lower sound fre-quencies (fsl) and at hypersound frequencies (fsh), this ratio yields a
similar value. The vibrational relaxation time of N2is larger than
10−4 seconds at room temperature,44 thus fslτvib ≫ 1 as well as
fshτvib≫ 1. In other words, the vibrational degrees remain frozen
under both conditions.45
For CO2, a different situation is encountered. At atmospheric
pressure, the vibrational relaxation time isτvib= 6× 10−6s, while
the rotational relaxation time isτrot = 3.8× 10−10 s (Refs.3and
46). Hence, for sound frequency measurements (at the megahertz scale),fslτvib≈ 1 and fslτrot≪ 1, which means that both rotation and
vibration are excited and take effect during energy exchange with translation in collisions. The method of sound absorption delivers an experimental ratio of bulk viscosity to shear viscosity (ηb/ηs) of
O(104) (Refs.42,47, and48). For hypersound frequencies (at the gigahertz scale),fshτvib≫ 1, causing the vibrational modes not to
take effect.
In order to describe macroscopic flow phenomena on time scales ranging from microseconds to nanoseconds, the used value of the bulk viscosity will range over four orders of magnitude. An example of such a flow phenomenon is a shock in a high Mach num-ber flow. The comparison of scattered light spectra to kinetic and hydrodynamic models in this paper shows that this dramatic fre-quency dependence of the bulk viscosity is due to the to the (gradual) cessation of vibrational relaxation.
ACKNOWLEDGMENTS
This research was supported by the China Exchange Program jointly run by the Netherlands Royal Academy of Sciences (KNAW) and the Chinese Ministry of Education. Y.W. acknowledges sup-port from the Chinese Scholarship Council (CSC) for his stay at VU Amsterdam. W.U. acknowledges the European Research Council for an ERC-Advanced grant under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 670168). The core part of the code that computes the Tenti models has been kindly provided to us by Xingguo Pan.
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